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Mirrors > Home > MPE Home > Th. List > ringacl | Structured version Visualization version GIF version |
Description: Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.) |
Ref | Expression |
---|---|
ringacl.b | ⊢ 𝐵 = (Base‘𝑅) |
ringacl.p | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
ringacl | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrp 19301 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | ringacl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | ringacl.p | . . 3 ⊢ + = (+g‘𝑅) | |
4 | 2, 3 | grpcl 18110 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1159 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Grpcgrp 18102 Ringcrg 19296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5209 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-ring 19298 |
This theorem is referenced by: ringcom 19328 ringlghm 19353 ringrghm 19354 imasring 19368 qusring2 19369 cntzsubr 19567 srngadd 19627 issrngd 19631 lmodprop2d 19695 prdslmodd 19740 psrlmod 20180 mpfind 20319 coe1add 20431 ip2subdi 20787 mat1ghm 21091 scmatghm 21141 mdetrlin2 21215 mdetunilem5 21224 cpmatacl 21323 mdegaddle 24667 deg1addle2 24695 deg1add 24696 ply1divex 24729 dvhlveclem 38243 baerlem3lem1 38842 mendlmod 39791 cznrng 44225 lmod1lem3 44543 |
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